Being interested in the mathematical theory, I was wondering if there are up-to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations).

It is clear that

in biological (population) models usually pregnancy introduces a delay term, or

in disease transition in networks the latent period introduces a delay, or

in engineering in feedback problems signal processing introduces the time delay.

I would like to see a list of answers where each answer contains one reference/example.

Rubber has the property that it takes time to adjust to the conditions it is applied to (Visco-Elasticity). The behaviour of these effects for static loads are known (Relaxation).

However with the high use of Rubbers for dynamic engineering applications (Example: Car tyres) it has been the new focus to study this property more thoroughly using dynamical loads and deriving a suitable Material Model (Lion & Höfer, Lion & Rendek).

The PDE's that arise from this are partly pure theoretical derived from a generalized Maxwell Modell combined with phenomenological effects.

In fluid dynamics it is often possible for a given geometry to isolate different degrees of freedom and model long range effects by a delayed influence. An example of this is the "delayed action oscillator" for the El Niño/Southern Oscillation phenomenon in oceanography.

Often fast degress of freedom are modeled by noise, i.e. the model consists of a delayed stochastic differential equation. An example for such a model can be seen here on the arXiv.

Besides the applications in climate science and abstract statistical physics, this kind of modelling is also important in engineering applications where the simplification is needed in order to get numerical results in time, for a successful control of a given system. Details can be found here:

I don't know much about the subject, except that it produces two interesting features that are not possible with local differential equations, mainly that a simple equation like this one

$$ \frac{dx}{dt} = b(t) x(t-1) $$

for suitable choices of the function $b$, this equation might have many initial states that produce the same final state, so they produce time asymmetric behavior automatically. Here is a more detailed, in depth exposition of the difficulties, specially regarding the insufficient initial data problem.

Another interesting result is that a classical harmonic oscillator with retardation will have a discrete spectrum of solutions, rather than a single one. Read this for some discussion.